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arXiv:2210.02402 · Physical Review A

Spin-based quantum Otto engines and majorization

Sachin Sonkar, Ramandeep S. Johal

Published Mar 31, 2023• 11 citations
arXivHTMLPDF

Abstract

The concept of majorization is explored as a tool to characterize the performance of a quantum Otto engine in the quasistatic regime. For a working substance in the form of a single spin of arbitrary magnitude, majorization yields a necessary and sufficient condition for the operation of the Otto engine, provided the canonical distribution of the working medium at the hot reservoir is majorized by its canonical distribution at the cold reservoir. We extend our analysis for two scenarios: a three-level atom and a bipartite system consisting of spin 1/2 interacting with an arbitrary spin via an isotropic Heisenberg exchange interaction. For both these cases, we find that while a majorization condition implies positive work extraction, it only yields sufficient conditions for the engine operation. Finally, we study the local thermodynamics of spins in the case of the bipartite system and infer an upper bound on the quantum Otto efficiency using the majorization relation.

DOI
10.1103/physreva.107.032220

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The concept of majorization is explored as a tool to characterize the performance of a quantum Otto engine in the quasistatic regime. For a working substance in the form of a single spin of arbitrary magnitude, majorization yields a necessary and sufficient condition for the operation of the Otto engine, provided the canonical distribution of the working medium at the hot reservoir is majorized by its canonical distribution at the cold reservoir. We extend our analysis for two scenarios: a three-level atom and a bipartite system consisting of spin 1/2 interacting with an arbitrary spin via an isotropic Heisenberg exchange interaction. For both these cases, we find that while a majorization condition implies positive work extraction, it only yields sufficient conditions for the engine operation. Finally, we study the local thermodynamics of spins in the case of the bipartite system and infer an upper bound on the quantum Otto efficiency using the majorization relation.
Quantum thermal machines make use of non-classical thermodynamic resources, one of which include interactions between elements of the quantum working medium. In this paper, we examine the performance of a quasi-static quantum Otto engine based on two spins of arbitrary magnitudes subject to an external magnetic field and coupled via an isotropic Heisenberg exchange interaction. It has been shown earlier that the said interaction provides an enhancement of cycle efficiency, with an upper bound that is tighter than the Carnot efficiency. However, the necessary conditions governing engine performance and the relevant upper bound for efficiency are unknown for the general case of arbitrary spin magnitudes. By analyzing extreme case scenarios, we formulate heuristics to infer the necessary conditions for an engine with uncoupled as well as coupled spin model. These conditions lead us to a connection between performance of quantum heat engines and the notion of majorization. Furthermore, the study of complete Otto cycles inherent in the average cycle also yields interesting insights into the average performance.
A three-level quantum system having two energy gaps presents a nontrivial working medium for a quantum heat engine. Our focus lies in understanding the constraints on the ability to modulate these gaps relative to the changes in probability distributions at the two given heat reservoirs. It is seen that an Otto engine in the quasistatic limit is feasible if at least one energy gap shrinks during the first quantum adiabatic stage. We analyze operating conditions under different variations of the gaps, revealing that a definite majorization relation between the hot and cold distributions serves as a sufficient criterion for the engine when both gaps are shrinking. Further, Otto efficiency is enhanced in case majorization holds. On the other hand, majorization becomes a necessary condition when only one of the gaps is shrinking. In the special case where one gap remains fixed, majorization is both necessary and sufficient for engine operation. For an nnn-level system, we note that a well defined change in energy gaps aligns with the majorization relation, thus characterizing the operation of the engine.
The performance of a quantum Otto engine is analyzed with regard to the constraints on the modulation of energy gaps relative to the changes in probability distributions at the two given heat reservoirs. We performed a detailed analysis with a generic three-level system (3LS), which serves as a non-trivial working medium with two energy gaps. A three-level Otto engine becomes feasible if at least one energy gap shrinks during the first quantum adiabatic stage. The operating regimes are derived for each allowed energy gap modulation, and majorization is observed to play a crucial role in determining the engine operation. This results in an enhanced Otto efficiency when the probability distributions fulfill the majorization condition. Finally, we show that our formalism applies to a swap engine based on a working medium composed of two 3LSs.
We consider a quasistatic quantum Otto cycle using two effectively two-level systems with degeneracy in the excited state. The systems are coupled through isotropic exchange interaction of strength J>0, in the presence of an external magnetic field B which is varied during the cycle. We prove the positive work condition and show that level degeneracy can act as a thermodynamic resource, so that a larger amount of work can be extracted than in the nondegenerate case, both with and without coupling. We also derive an upper bound for the efficiency of the cycle. This bound is the same as derived for a system of coupled spin-1/2 particles [G. Thomas and R. S. Johal, Phys. Rev. E 83, 031135 (2011)PLEEE81539-375510.1103/PhysRevE.83.031135], i.e., without degeneracy, and depends only on the control parameters of the Hamiltonian, being independent of the level degeneracy and the reservoir temperatures.
We investigate a quantum heat engine with a working substance of two particles, one with a spin-1/2 and the other with an arbitrary spin (spin s), coupled by Heisenberg exchange interaction, and subject to an external magnetic field. The engine operates in a quantum Otto cycle. Work harvested in the cycle and its efficiency are calculated using quantum thermodynamical definitions. It is found that the engine has higher efficiencies at higher spins and can harvest work at higher exchange interaction strengths. The role of exchange coupling and spin s on the work output and the thermal efficiency is studied in detail. In addition, the engine operation is analyzed from the perspective of local work and efficiency. We develop a general formalism to explore local thermodynamics applicable to any coupled bipartite system. Our general framework allows for examination of local thermodynamics even when global parameters of the system are varied in thermodynamic cycles. The generalized definitions of local and cooperative work are introduced by using mean field Hamiltonians. The general conditions for which the global work is not equal to the sum of the local works are given in terms of the covariance of the subsystems. Our coupled spin quantum Otto engine is used as an example of the general formalism.
Quantum Otto and Carnot engines have recently been receiving attention due to their ability to achieve high efficiencies and powers based on the laws of quantum mechanics. This paper discusses the theory, progress, and possible applications of quantum Otto and Carnot engines, such as energy production, cooling, and nanoscale technologies. In particular, we investigate a two-spin Heisenberg system that works as a substance in quantum Otto and Carnot cycles while exposed to an external magnetic field with both Dzyaloshinsky-Moriya and dipole-dipole interactions. The four stages of engine cycles are subject to analysis with respect to the heat exchanges that occur between the hot and cold reservoirs, alongside the work done during each stage. The operating conditions of the heat engine, refrigerator, thermal accelerator, and heater are all achieved. Moreover, our results demonstrate that the laws of thermodynamics are strictly upheld and the Carnot cycle produces more useful work than that of the Otto cycle.